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\begin{document}
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	\begin{align*}
		&\ \lim_{x \to \infty} (\frac{a x + b}{a x + c})^{h x + k} \ \ (a \neq 0)
		\\
		= &\ \lim_{x \to \infty} e^{(h x + k) \ln{(\frac{a x + b}{a x + c})}}
		\\
		= &\ \lim_{x \to \infty} e^{(h x + k) \ln{(\frac{a x + b + c - c}{a x + c})}}
		\\
		= &\ \lim_{x \to \infty} e^{(h x + k) \ln{(1 + \frac{b - c}{a x + c})}}
		\\
		= &\ \lim_{x \to \infty} e^{(h x + k) \frac{b - c}{a x + c}}
		\\
		= &\ \lim_{x \to \infty} e^{\frac{(b - c) h x + (b - c) k}{a x + c}}
		\\
		= &\ e^{\frac{(b - c) h}{a}}
	\end{align*}
\end{document}
